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Properties

Label 798.2.bu.b
Level $798$
Weight $2$
Character orbit 798.bu
Analytic conductor $6.372$
Analytic rank $0$
Dimension $162$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [798,2,Mod(53,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("798.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 12, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bu (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [162,0,0,0,0,0,0,81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(27\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 162 q + 81 q^{8} + 15 q^{13} - 6 q^{14} + 3 q^{15} + 6 q^{17} - 18 q^{18} + 9 q^{19} - 6 q^{22} - 6 q^{25} + 9 q^{27} + 9 q^{28} - 6 q^{29} + 24 q^{33} + 6 q^{34} - 24 q^{35} + 18 q^{37} - 9 q^{39} - 9 q^{43}+ \cdots + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −0.173648 − 0.984808i −1.72540 + 0.151635i −0.939693 + 0.342020i −1.01302 + 2.78324i 0.448944 + 1.67286i −1.46179 + 2.20526i 0.500000 + 0.866025i 2.95401 − 0.523261i 2.91686 + 0.514321i
53.2 −0.173648 − 0.984808i −1.70606 − 0.298913i −0.939693 + 0.342020i 1.15887 − 3.18398i 0.00188265 + 1.73205i 0.0260849 − 2.64562i 0.500000 + 0.866025i 2.82130 + 1.01993i −3.33685 − 0.588376i
53.3 −0.173648 − 0.984808i −1.70114 + 0.325790i −0.939693 + 0.342020i −0.683985 + 1.87923i 0.616239 + 1.61872i 0.807718 − 2.51944i 0.500000 + 0.866025i 2.78772 − 1.10842i 1.96946 + 0.347268i
53.4 −0.173648 − 0.984808i −1.60746 + 0.645045i −0.939693 + 0.342020i −0.0712889 + 0.195865i 0.914377 + 1.47102i −2.34853 + 1.21837i 0.500000 + 0.866025i 2.16783 − 2.07376i 0.205268 + 0.0361943i
53.5 −0.173648 − 0.984808i −1.50443 + 0.858307i −0.939693 + 0.342020i 1.35069 − 3.71100i 1.10651 + 1.33253i 0.939767 + 2.47322i 0.500000 + 0.866025i 1.52662 − 2.58253i −3.88916 − 0.685764i
53.6 −0.173648 − 0.984808i −1.47865 − 0.901990i −0.939693 + 0.342020i 0.639343 − 1.75658i −0.631522 + 1.61282i 1.48103 + 2.19239i 0.500000 + 0.866025i 1.37283 + 2.66746i −1.84091 − 0.324603i
53.7 −0.173648 − 0.984808i −1.29554 − 1.14959i −0.939693 + 0.342020i −0.752515 + 2.06752i −0.907159 + 1.47549i −1.40821 − 2.23985i 0.500000 + 0.866025i 0.356873 + 2.97870i 2.16678 + 0.382062i
53.8 −0.173648 − 0.984808i −1.02584 + 1.39558i −0.939693 + 0.342020i 0.359967 − 0.989002i 1.55252 + 0.767912i −2.21348 − 1.44931i 0.500000 + 0.866025i −0.895314 − 2.86329i −1.03648 − 0.182760i
53.9 −0.173648 − 0.984808i −1.02304 − 1.39764i −0.939693 + 0.342020i 0.308773 − 0.848347i −1.19875 + 1.25020i 2.64567 + 0.0204696i 0.500000 + 0.866025i −0.906776 + 2.85968i −0.889077 − 0.156768i
53.10 −0.173648 − 0.984808i −0.822888 + 1.52409i −0.939693 + 0.342020i −1.38704 + 3.81087i 1.64383 + 0.545731i 2.01440 + 1.71528i 0.500000 + 0.866025i −1.64571 − 2.50831i 3.99383 + 0.704221i
53.11 −0.173648 − 0.984808i −0.801885 + 1.53525i −0.939693 + 0.342020i −0.146667 + 0.402963i 1.65117 + 0.523110i 2.63845 + 0.196355i 0.500000 + 0.866025i −1.71396 − 2.46218i 0.422310 + 0.0744646i
53.12 −0.173648 − 0.984808i −0.550472 − 1.64225i −0.939693 + 0.342020i −0.426546 + 1.17193i −1.52171 + 0.827283i −0.140289 + 2.64203i 0.500000 + 0.866025i −2.39396 + 1.80803i 1.22819 + 0.216563i
53.13 −0.173648 − 0.984808i −0.0720082 + 1.73055i −0.939693 + 0.342020i −0.564194 + 1.55011i 1.71677 − 0.229593i −0.631754 − 2.56922i 0.500000 + 0.866025i −2.98963 − 0.249228i 1.62453 + 0.286449i
53.14 −0.173648 − 0.984808i 0.0926672 − 1.72957i −0.939693 + 0.342020i 1.09478 − 3.00789i −1.71939 + 0.209077i −1.00516 − 2.44738i 0.500000 + 0.866025i −2.98283 − 0.320549i −3.15230 − 0.555835i
53.15 −0.173648 − 0.984808i 0.141707 − 1.72624i −0.939693 + 0.342020i −1.13790 + 3.12636i −1.72463 + 0.160205i 2.35336 − 1.20900i 0.500000 + 0.866025i −2.95984 − 0.489242i 3.27646 + 0.577729i
53.16 −0.173648 − 0.984808i 0.785925 + 1.54348i −0.939693 + 0.342020i 0.859500 − 2.36146i 1.38355 − 1.04201i 1.57496 − 2.12591i 0.500000 + 0.866025i −1.76464 + 2.42611i −2.47483 − 0.436380i
53.17 −0.173648 − 0.984808i 0.826628 − 1.52207i −0.939693 + 0.342020i 0.571310 − 1.56966i −1.64249 − 0.549766i 2.51710 − 0.815004i 0.500000 + 0.866025i −1.63337 − 2.51637i −1.64502 − 0.290062i
53.18 −0.173648 − 0.984808i 0.912860 − 1.47197i −0.939693 + 0.342020i −0.436242 + 1.19856i −1.60812 − 0.643387i −0.993792 + 2.45201i 0.500000 + 0.866025i −1.33337 − 2.68740i 1.25611 + 0.221486i
53.19 −0.173648 − 0.984808i 0.994724 + 1.41793i −0.939693 + 0.342020i 1.36048 − 3.73789i 1.22366 − 1.22583i −2.59662 − 0.507500i 0.500000 + 0.866025i −1.02105 + 2.82090i −3.91734 − 0.690734i
53.20 −0.173648 − 0.984808i 1.07679 + 1.35666i −0.939693 + 0.342020i −0.179971 + 0.494467i 1.14907 − 1.29601i 1.75103 + 1.98341i 0.500000 + 0.866025i −0.681063 + 2.92167i 0.518206 + 0.0913737i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.27
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
399.br even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bu.b yes 162
3.b odd 2 1 798.2.bu.a ✓ 162
7.c even 3 1 798.2.cc.b yes 162
19.f odd 18 1 798.2.cc.a yes 162
21.h odd 6 1 798.2.cc.a yes 162
57.j even 18 1 798.2.cc.b yes 162
133.be odd 18 1 798.2.bu.a ✓ 162
399.br even 18 1 inner 798.2.bu.b yes 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bu.a ✓ 162 3.b odd 2 1
798.2.bu.a ✓ 162 133.be odd 18 1
798.2.bu.b yes 162 1.a even 1 1 trivial
798.2.bu.b yes 162 399.br even 18 1 inner
798.2.cc.a yes 162 19.f odd 18 1
798.2.cc.a yes 162 21.h odd 6 1
798.2.cc.b yes 162 7.c even 3 1
798.2.cc.b yes 162 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{162} + 3 T_{5}^{160} - 18 T_{5}^{159} + 27 T_{5}^{158} + 12 T_{5}^{157} - 13220 T_{5}^{156} + \cdots + 56\!\cdots\!68 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display